Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userLie antialgebras: premices
202
0
0.0
(
0
)
Authors
Valentin Ovsienko
Ask ChatGPT about the research
No Arabic abstract
The main purpose of this work is to develop the basic notions of the Lie theory for commutative algebras. We introduce a class of $mathbbZ_2$-graded commutative but not associative algebras that we call ``Lie antialgebras. These algebras are closely related to Lie (super)algebras and, in some sense, link together commutative and Lie algebras. The main notions we define in this paper are: representations of Lie antialgebras, an analog of the Lie-Poisson bivector (which is not Poisson) and central extensions. We also classify simple finite-dimensional Lie antialgebras.
rate research
Read More
Rota-Baxter algebras were introduced to solve some analytic and combinatorial problems and have appeared in many fields in mathematics and mathematical physics. Rota-Baxter algebras provide a construction of pre-Lie algebras from associative algebras. In this paper, we give all Rota-Baxter operators of weight 1 on complex associative algebras in dimension $leq 3$ and their corresponding pre-Lie algebras.
We study solutions of the Bethe ansatz equations associated to the orthosymplectic Lie superalgebras $mathfrak{osp}_{2m+1|2n}$ and $mathfrak{osp}_{2m|2n}$. Given a solution, we define a reproduction procedure and use it to construct a family of new solutions which we call a population. To each population we associate a symmetric rational pseudo-differential operator $mathcal R$. Under some technical assumptions, we show that the superkernel $W$ of $mathcal R$ is a self-dual superspace of rational functions, and the population is in a canonical bijection with the variety of isotropic full superflags in $W$ and with the set of symmetric complete factorizations of $mathcal R$. In particular, our results apply to the case of even Lie algebras of type D${}_m$ corresponding to $mathfrak{osp}_{2m|0}=mathfrak{so}_{2m}$.
In this paper, we first construct the controlling algebras of embedding tensors and Lie-Leibniz triples, which turn out to be a graded Lie algebra and an $L_infty$-algebra respectively. Then we introduce representations and cohomologies of embedding tensors and Lie-Leibniz triples, and show that there is a long exact sequence connecting various cohomologies. As applications, we classify infinitesimal deformations and central extensions using the second cohomology groups. Finally, we introduce the notion of a homotopy embedding tensor which will induce a Leibniz$_infty$-algebra. We realize Kotov and Strobls construction of an $L_infty$-algebra from an embedding tensor, to a functor from the category of homotopy embedding tensors to that of Leibniz$_infty$-algebras, and a functor further to that of $L_infty$-algebras.
In this paper, first we introduce the notion of a Reynolds operator on an $n$-Lie algebra and illustrate the relationship between Reynolds operators and derivations on an $n$-Lie algebra. We give the cohomology theory of Reynolds operators on an $n$-Lie algebra and study infinitesimal deformations of Reynolds operators using the second cohomology group. Then we introduce the notion of NS-$n$-Lie algebras, which are generalizations of both $n$-Lie algebras and $n$-pre-Lie algebras. We show that an NS-$n$-Lie algebra gives rise to an $n$-Lie algebra together with a representation on itself. Reynolds operators and Nijenhuis operators on an $n$-Lie algebra naturally induce NS-$n$-Lie algebra structures. Finally, we construct Reynolds $(n+1)$-Lie algebras and Reynolds $3$-Lie algebras from Reynolds $n$-Lie algebras and Reynolds commutative associative algebras respectively.
In this thesis new objects to the existing set of invariants of Lie algebras are added. These invariant characteristics are capable of describing the nilpotent parametric continuum of Lie algebras. The properties of these invariants, in view of possible alternative classifications of Lie algebras, are formulated and their behaviour on known lower--dimensional Lie algebras investigated. It is demonstrated that these invariants, in view of their application on graded contractions of sl(3,C), are also effective in higher dimensions. A necessary contraction criterion involving these invariants is derived and applied to lower--dimensional cases. Possible application of these invariant characteristics to Jordan algebras is investigated.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
